Let D be a bounded domain in an n-dimensional Euclidean space ? n . Assume that $$0 < \lambda _1 \leqslant \lambda _2 \leqslant \cdots \leqslant \lambda _k \leqslant \cdots $$ are the eigenvalues of the Dirichlet L...Let D be a bounded domain in an n-dimensional Euclidean space ? n . Assume that $$0 < \lambda _1 \leqslant \lambda _2 \leqslant \cdots \leqslant \lambda _k \leqslant \cdots $$ are the eigenvalues of the Dirichlet Laplacian operator with any order l: $$\left\{ \begin{gathered} ( - \vartriangle )^l u = \lambda u, in D \hfill \\ u = \frac{{\partial u}}{{\partial \vec n}} = \cdots = \frac{{\partial ^{l - 1} u}}{{\partial \vec n^{l - 1} }} = 0, on \partial D \hfill \\ \end{gathered} \right.$$ . Then we obtain an upper bound of the (k+1)-th eigenvalue λ k+1 in terms of the first k eigenvalues. $$\sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )} \leqslant \frac{1}{n}[4l(n + 2l - 2)]^{\tfrac{1}{2}} \left\{ {\sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )^{\tfrac{1}{2}} \lambda _i^{\tfrac{{l - 1}}{l}} \sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )^{\tfrac{1}{2}} \lambda _i^{\tfrac{1}{l}} } } } \right\}^{\tfrac{1}{2}} $$ . This ineguality is independent of the domain D. Furthermore, for any l ? 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang.展开更多
In this paper, we study Lichnerowicz type estimate for eigenvalues of drifting Laplacian operator and the decay rates of L1 and L2 energy for drifting heat equation on closed Riemannian manifolds with weighted measure.
This paper deals with the discrete-time connected coverage problem with the constraint that only local information can be utilized for each robot. In such distributed framework, global connectivity characterized by th...This paper deals with the discrete-time connected coverage problem with the constraint that only local information can be utilized for each robot. In such distributed framework, global connectivity characterized by the second smallest eigenvalue of topology Laplacian is estimated through introducing distributed minimal-time consensus algorithm and power iteration algorithm. A self-deployment algorithm is developed to disperse the robots with the precondition that the estimated second smallest eigenvalue is positive at each time-step. Since thus connectivity constraint does not impose to preserve some certain edges, the self-deployment strategy developed in this paper reserves a sufficient degree of freedom for the motion of robots. Theoretical analysis demonstrates that each pair of neighbor robots can finally reach the largest objective distance from each other while the group keeps connected all the time, which is also shown by simulations.展开更多
In this paper we consider the Dirichlet problem■where p is a small parameter andΩis a C^(2)bounded domain in R^(2).In[1],the author proves the existence of a m-point blow-up solution u_(p)jointly with its asymptotic...In this paper we consider the Dirichlet problem■where p is a small parameter andΩis a C^(2)bounded domain in R^(2).In[1],the author proves the existence of a m-point blow-up solution u_(p)jointly with its asymptotic behaviour.We compute the Morse index of up in terms of the Morse index of the associated Hamilton function of this problem.In addition,we give an asymptotic estimate for the first 4m eigenvalues and eigenfunctions.展开更多
基金the National Natural Science Foundation of China(Grant No.10571088)
文摘Let D be a bounded domain in an n-dimensional Euclidean space ? n . Assume that $$0 < \lambda _1 \leqslant \lambda _2 \leqslant \cdots \leqslant \lambda _k \leqslant \cdots $$ are the eigenvalues of the Dirichlet Laplacian operator with any order l: $$\left\{ \begin{gathered} ( - \vartriangle )^l u = \lambda u, in D \hfill \\ u = \frac{{\partial u}}{{\partial \vec n}} = \cdots = \frac{{\partial ^{l - 1} u}}{{\partial \vec n^{l - 1} }} = 0, on \partial D \hfill \\ \end{gathered} \right.$$ . Then we obtain an upper bound of the (k+1)-th eigenvalue λ k+1 in terms of the first k eigenvalues. $$\sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )} \leqslant \frac{1}{n}[4l(n + 2l - 2)]^{\tfrac{1}{2}} \left\{ {\sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )^{\tfrac{1}{2}} \lambda _i^{\tfrac{{l - 1}}{l}} \sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )^{\tfrac{1}{2}} \lambda _i^{\tfrac{1}{l}} } } } \right\}^{\tfrac{1}{2}} $$ . This ineguality is independent of the domain D. Furthermore, for any l ? 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang.
基金Supported by National Natural Science Foundation of China(Grant No.11271111)
文摘In this paper, we study Lichnerowicz type estimate for eigenvalues of drifting Laplacian operator and the decay rates of L1 and L2 energy for drifting heat equation on closed Riemannian manifolds with weighted measure.
基金the National Natural Science Foundation of China(Nos.61203073 and 61271114)the Specialized Research Fund for the Doctoral Program of Higher Education of China(No.20120075120008)the Foundation of Key Laboratory of System Control and Information Processing,Ministry of Education,China(No.SCIP2012002)
文摘This paper deals with the discrete-time connected coverage problem with the constraint that only local information can be utilized for each robot. In such distributed framework, global connectivity characterized by the second smallest eigenvalue of topology Laplacian is estimated through introducing distributed minimal-time consensus algorithm and power iteration algorithm. A self-deployment algorithm is developed to disperse the robots with the precondition that the estimated second smallest eigenvalue is positive at each time-step. Since thus connectivity constraint does not impose to preserve some certain edges, the self-deployment strategy developed in this paper reserves a sufficient degree of freedom for the motion of robots. Theoretical analysis demonstrates that each pair of neighbor robots can finally reach the largest objective distance from each other while the group keeps connected all the time, which is also shown by simulations.
文摘In this paper we consider the Dirichlet problem■where p is a small parameter andΩis a C^(2)bounded domain in R^(2).In[1],the author proves the existence of a m-point blow-up solution u_(p)jointly with its asymptotic behaviour.We compute the Morse index of up in terms of the Morse index of the associated Hamilton function of this problem.In addition,we give an asymptotic estimate for the first 4m eigenvalues and eigenfunctions.
